Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion. II: Global structure (Q865276)

This paper is concerned with the global bifurcation structure of positive solutions for the following stationary problem: \[ {\mathbf{0}}=\epsilon D{\mathbf{u}}''+{\widetilde{\mathbf{f}}} ({\mathbf{u}})\text{ for } x\in(0,\pi), \] \[ u'(0)=u'(\pi)=0 \] of a Lotka-Volterra competition model with diffusion, where \(D=\text{diag}(d_u,d_v)\), \[ {\mathbf{u}}=(u,v),{\widetilde{\mathbf{f}}}=({\widetilde{f}}, {\widetilde{g}}),\;{\widetilde{f}}({\mathbf{u}})= {\widetilde{f}}^0({\mathbf{u}})u,{\widetilde{g}}({\mathbf{u}})= {\widetilde{g}}^0({\mathbf{u}})v. \] It is shown that the global bifurcation structure of positive stationary solutions for the above model is similar to that for a certain scalar reaction-diffusion equation. To do this, the comparison principle, bifurcation theory and numerical verification are employed. Part I, cf. J. Comput. Appl. Math. 201, No. 2, 317--326 (2007; Zbl 1110.65067).